To demonstrate the effectiveness of the shape parameter update rule (REF ), we compare the performance of the conventional SBL algorithm with and without shape parameter update. The results are shown in Fig. REF , where the SNR is 50dB, the size of the measurement matrix is \(800 \times 1000\) , and the sparsity rate \(\rho = 0.1\) . In this figure, the support-oracle bound is also shown for reference. The matrices in (a), (b), and (c) are respectively i.i.d. Gaussian, correlated and low-rank matrices (refer to blackSection VI for their generations). It can be seen that there is a clear gap between the performance of the conventional SBL and the bounds, and with shape parameter updated with our rule, the SBL algorithm attains the bound. It is worth mentioning the empirical finding in [1]}, i.e., replacing the noise variance \(\beta ^{-1}\)  with \(3 \beta ^{-1}\)  can lead to better performance of GGAMP-SBL [1]}. We use this for the conventional SBL algorithm and the performance is also included in Fig. REF . blackWe see that it also leads to substantial performance improvement, but its performance is inferior to that of SBL with updated \(\epsilon \)  using (REF ). Moreover, in many cases, the noise variance is unknown, and it may be hard to determine its value accurately. In contrast, our empirical update of \(\epsilon \)  does not require any additional information.
